An Analysis to Treat the Degeneracy of a Basic Feasible Solution in Interval Linear Programming

• Published in: Integrated Uncertainty in Knowledge Modelling and Decision Making Vol. 13199; pp. 130 - 142
• Main Authors: Gao, Zhenzhong, Inuiguchi, Masahiro
• Format: Book Chapter
• Language: English
• Published: Switzerland Springer International Publishing AG 2022; Springer International Publishing
• Series: Lecture Notes in Computer Science
• Subjects: Basic space; Degeneracy; Interval linear programming; Polyhedral convex cone; Tangent cone
• ISBN: 3030980170; 9783030980177
• ISSN: 0302-9743; 1611-3349
• DOI: 10.1007/978-3-030-98018-4_11

When coefficients in the objective function cannot be precisely determined, the optimal solution is fluctuated by the realisation of coefficients. Therefore, analysing the stability of an optimal solution becomes essential. Although the robustness analysis of an optimal basic solution has been developed successfully so far, it becomes complex when the solution contains degeneracy. This study is devoted to overcoming the difficulty caused by the degeneracy in a linear programming problem with interval objective coefficients. We focus on the tangent cone of a degenerate basic feasible solution since the belongingness of the objective coefficient vector to its associated normal cone assures the solution’s optimality. We decompose the normal cone by its associated tangent cone to a direct union of subspaces. Several propositions related to the proposed approach are given. To demonstrate the significance of the decomposition, we consider the case where the dimension of the subspace is one. We examine the obtained propositions by numerical examples with comparisons to the conventional techniques.